Simple strategies that generate bounded solutions for the multiple‐choice multi‐dimensional knapsack problem: a guide for OR practitioners

Abstract

The multiple‐choice multi‐dimensional knapsack problem (MMKP) is an NP‐hard generalization of the classic 0–1 knapsack problem. The MMKP has a variety of important industrial and business applications. Approximate solution approaches characteristically give no guarantees on solution quality. Exact solution approaches usually generate solutions that are guaranteed to be close to the optimum but typically take excessive computation times—one to several hours have been recorded in the literature. In this article, we use both simple single‐step and multiple step strategies to demonstrate how a commercial software package (Gurobi in this case) can easily be used to generate guaranteed tightly bounded solutions for 293 MMKP instances commonly tested in the literature. These four simple strategies that require no problem‐specific algorithms are shown to perform competitively with the five best published algorithms for the MMKP. Since no problem‐specific coding is required, these simple strategies are easy for operations research practitioners to use.

Using general-purpose integer programming software to generate bounded solutions for the multiple knapsack problem: a guide for or practitioners

“…Trying to have the best of two worlds (a method that is both fast and guarantees solution quality) is the motivation [13] behind the SSIT algorithm. More benefits of SSIT are detailed in [19]. Successful applications of SSIT to solve several binary integer programs (BIP) have been documented in the literature [14][15][16][17][18][19].…”

“…More benefits of SSIT are detailed in [19]. Successful applications of SSIT to solve several binary integer programs (BIP) have been documented in the literature [14][15][16][17][18][19]. For these applications, SSIT typically generates solutions guaranteed to be within 0.1% of the optimums in about 60 seconds on standard PCs.…”

Modified Artificial Bee Colony Algorithm for Multiple-Choice Multidimensional Knapsack Problem

“…Lamanna et al [74] provided a new variant of the heuristic framework kernel search applied to the MMKP. Dellinger et al [75] proposed simple strategies that generate bounded solutions for the MMKP.…”

LP relaxation and dynamic programming enhancing VNS for the multiple knapsack problem with setup

“…The KP (Martello and Toth, 1990; Kellerer et al., 2004; Cacchiani et al., 2022a) is a kind of a well‐known non‐deterministic polynomial time‐hard hard (NP‐hard) roblem and a typical combinatorial optimization problem, which has more vital irreplaceable practical value in many yields such as resource allocations, capital budgets, investment decisions, and industrial loading. There exist many extensions of the KP including the multi‐dimensional KP (Kellerer et al., 2004; Wang and Yichao, 2017; Cacchiani et al., 2022b), the multiple‐choice KP (Lin, 1998), the multiple KP (Kellerer et al., 2004; Dell'Amico et al., 2019; Sur et al., 2022), the multi‐demand multi‐dimensional KP (Lai et al., 2019), the multiple‐choice multi‐dimensional KP (Dellinger et al., 2022), the max‐min KP (Wang and Yichao, 2017), the quadratic multiple KP (Aïder et al., 2022; Fleszar, 2022), the generalized quadratic multiple KP (Saraç and Sipahioglu, 2014; Avci and Topaloglu, 2017; Adouani et al., 2019), the KPS (Chebil and Khemakhem, 2015; Della et al., 2017; Furini et al., 2018; Amri, 2019; Boukhari et al., 2020), the MKPS (Yang, 2006; Lahyani et al., 2019; Adouani et al., 2020; Amiri and Barkhi, 2020; Boukhari et al., 2022a), and the generalized MKPS (Adouani et al., 2020; Boukhari et al., 2022b).…”